Symmetries and exact solutions of fractional filtration equations

Газизов, Р. К.; Kasatkin, Alexey A.; Lukashchuk, Stanislav Yu.

doi:10.1063/1.5012621

Cited by 11 publications

(20 citation statements)

References 17 publications

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“…is the total derivative. Gazizov et al in [5] examined the transformation of the form (5) that conserve the structure of the fractional derivative operator (3) and the infinitesimal transformation of fractional derivative is introduced:…”

Section: Preliminariesmentioning

confidence: 99%

“…The application of Lie group analysis to fractional calculus, on the other hand, is relatively underrated. Gazizov et al, in 2007, formulated the prolongation formulae for fractional derivative [5]. Huang and Zhdanov then gave an explicit form of the finding in [5] in their paper later in 2014 [6].…”

Section: Introductionmentioning

confidence: 99%

“…Gazizov et al, in 2007, formulated the prolongation formulae for fractional derivative [5]. Huang and Zhdanov then gave an explicit form of the finding in [5] in their paper later in 2014 [6]. The methods of Lie have been applied to linear and nonlinear partial differential equation with some success, for example see [11,12,13].…”

Section: Introductionmentioning

confidence: 99%

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Lie symmetry analysis of a fractional Black-Scholes equation

Chong¹,

O’Hara²

2019

AIP Conference Proceedings

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Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Lie symmetry analysis of a fractional Black-Scholes equation

Chong¹,

O’Hara²

2019

AIP Conference Proceedings

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“…Caputo [9] introduced a memory formalism using the apparatus of fractional calculus and proposed the time fractional modification of the classical Darcy's law. This idea was further developed by many authors (e.g., [6,7,[10][11][12][13][14]), and a number of fractional differential generalizations of the classical model have been developed that take into account the memory and spatial correlations effects in porous media (see a literature review in [15]). In this case, the order of the fractional derivative determines the degree of memory impact on patterns of the flow.…”

Section: Introductionmentioning

confidence: 99%

Convergence Analysis of a Numerical Method for a Fractional Model of Fluid Flow in Fractured Porous Media

2021

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The present paper is devoted to the construction and study of numerical methods for solving an initial boundary value problem for a differential equation containing several terms with fractional time derivatives in the sense of Caputo. This equation is suitable for describing the process of fluid flow in fractured porous media under some physical assumptions, and has an important applied significance in petroleum engineering. Two different approaches to constructing numerical schemes depending on orders of the fractional derivatives are proposed. The semi-discrete and fully discrete numerical schemes for solving the problem are analyzed. The construction of a fully discrete scheme is based on applying the finite difference approximation to time derivatives and the finite element method in the spatial direction. The approximation of the fractional derivatives in the sense of Caputo is carried out using the L1-method. The convergence of both numerical schemes is rigorously proved. The results of numerical tests conducted for model problems are provided to confirm the theoretical analysis. In addition, the proposed computational method is applied to study the flow of oil in a fractured porous medium within the framework of the considered model. Based on the results of the numerical tests, it was concluded that the model reproduces the characteristic features of the fluid flow process in the medium under consideration.

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“…In the last few decades, the subject of the fractional calculus has caught the consideration of many researchers, who contributed to its development. Recently, some analytical and numerical methods [3,8] have been introduced to solve a fractional order differential equation. However, all the methods have an insufficient development as they allow one to find solutions only in case of linear equations and for isolated examples of nonlinear equations [1,[15][16][17][18][19].…”

Section: Introductionmentioning

confidence: 99%

Symmetry Classification and Exact Solutions of a Variable Coefficient Space-Time Fractional Potential Burgers’ Equation

Gaur

Singh

2016

International Journal of Differential Equations

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In this paper, we investigate the symmetry properties of a variable coefficient nonlinear space-time fractional Burgers' equation. Fractional Lie symmetries and corresponding infinitesimal generators are obtained. With the help of the infinitesimal generators some group invariant solutions are deduced. Further, some exact solutions of fractional Burgers' equation are generated by the invariant subspace method. MSC 2010: Primary 26A33: Secondary 33E12, 34A08, 34K37, 35R11, 60G22

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Symmetries and exact solutions of fractional filtration equations

Cited by 11 publications

References 17 publications

Lie symmetry analysis of a fractional Black-Scholes equation

Lie symmetry analysis of a fractional Black-Scholes equation

Convergence Analysis of a Numerical Method for a Fractional Model of Fluid Flow in Fractured Porous Media

Symmetry Classification and Exact Solutions of a Variable Coefficient Space-Time Fractional Potential Burgers’ Equation

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